The confluent hypergeometric differential equation
has a regular singular point at
and an essential singularity at
. Solutions analytic at
are confluent hypergeometric functions of the first kind (or Kummer functions):
are Pochhammer symbols defined by
, the function becomes singular, unless
is an equal or smaller negative integer (
), and it is convenient to define the regularized confluent hypergeometric
, which is an entire function for all values of
The second, linearly independent solutions of the differential equation are confluent hypergeometric functions of the second kind (or Tricomi functions), defined by
, where the generalized hypergeometric function
represents a formal asymptotic series.
If the hypergeometric function with argument
is complex, both the real and imaginary parts are plotted (black and red curves).
For certain choices of the parameters
, the hypergeometric functions are related to various transcendental and special functions. Several illustrations are given in the snapshots.